38 STATEMENT AND EXPOSITION OF 



All this, so far, approximates to an accordance with M. Le Verrier's required 

 action of the mass in question. It is then sufficiently manifest that our hypothetical 

 planet, as to mass and distance both, would be such as measurably to satisfy the 

 conditions of the ascertained perturbation ; and so we need not pursue the investi- 

 gation of a troublesome problem any farther. 



Pecidiar Relations of the Living Force of {siimdtaneotis) Rotation of some of the 

 Planetary and Satellite Masses. 



(53) If Jupiter and Saturn should (or if they did) turn around the sun, in the 

 same time ; the moment of rotation must, in the instance of either, be represented 

 by the formula, mass x {velocityf ; or, as velocity in this case would be, as a, the 

 radius vector of rotation, the ratio of the moments will be obtained by comparing 

 Tuass X (radius vectorY of the one with mass x (radius vectorf of the other. So 

 with tn and m', respectively, for the masses, and a and d for the radii vectores ; i. e. 

 the mean distances from the sun, as in the column of Law in Table (B\ in (14), 

 and the masses, as in Table (J.), in (3) ; we have — 



For Jupiter, ma"" = 0.026142. 

 For Saturn, ??('a'- = 0.025477. 



or with the distances as in column of Fact in Table (B) ; we have — 



For Jupiter, ma'' = 0.025832. 

 For Saturn, m'a'^= 0.025985. 



The approach to a ratio of equality is here very close.^ 



There is also an approximation to the same state of things in the following 

 cases.^ 



The respective moments of (simultaneous) rotation of \^ (i. e. Saturn reduced to 

 its ancient state), of Uranus, and also of 6i [the half-planet (supplied) interior to 

 Uranus], are all nearly equal to one another ; the ratios being — 



1.1431 (1). 



1.0060 (2). 





mr 



'^i' 



Then, when the comhined masses of Saturn and Uranus [in the More Ancient 

 State, as exhibited in the term [(U)>^], in Table (F), in (45)], are compared with 

 Neptime in respect to the moment of (simultaneous) rotation; we have for the 

 ratio — 



' This curious relation was first made known by the author of this paper to the American Associa- 

 tion for the Advancement of Science, at their Meeting in Montreal, in 185Y ; also the division into 

 shells, etc. 



' Which might be somewhat varied, were all the masses more accurately determined. 



